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About posets of height one as retracts

Frank a Campo

Abstract

We investigate connected posets $C$ of height one as retracts of finite posets $P$. We define two multigraphs: a multigraph $\mathfrak{F}(P)$ reflecting the network of so-called improper 4-crown bundles contained in the extremal points of $P$, and a multigraph $\mathfrak{C}(C)$ depending on $C$ but not on $P$. There exists a close interdependence between $C$ being a retract of $P$ and the existence of a graph homomorphism of a certain type from $\mathfrak{F}(P)$ to $\mathfrak{C}(C)$. In particular, if $C$ is an ordinal sum of two antichains, then $C$ is a retract of $P$ iff such a graph homomorphism exists. Returning to general connected posets $C$ of height one, we show that the image of such a graph homomorphism can be a clique in $\mathfrak{C}(C)$ iff the improper 4-crowns in $P$ contain only a sparse subset of the edges of $C$.

About posets of height one as retracts

Abstract

We investigate connected posets of height one as retracts of finite posets . We define two multigraphs: a multigraph reflecting the network of so-called improper 4-crown bundles contained in the extremal points of , and a multigraph depending on but not on . There exists a close interdependence between being a retract of and the existence of a graph homomorphism of a certain type from to . In particular, if is an ordinal sum of two antichains, then is a retract of iff such a graph homomorphism exists. Returning to general connected posets of height one, we show that the image of such a graph homomorphism can be a clique in iff the improper 4-crowns in contain only a sparse subset of the edges of .

Paper Structure

This paper contains 8 sections, 9 theorems, 25 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preparation
  3. Notation and basic terms
  4. Results about homomorphisms
  5. Posets of height one as retracts
  6. The multigraphs $\mathfrak{ C }(C)$ and $\mathfrak{ F }(P)$
  7. Separating homomorphisms and retracts
  8. Cliques as images of homomorphisms

Key Result

Lemma 1

Let $f : P \rightarrow Q$ be a surjective homomorphism. There exists a homomorphism $g : P \rightarrow Q$ with $g[L(P)] = L(Q)$, $g[U(P)] = U(Q)$, and $g \vert_{M(P)} = f \vert_{M(P)}$. If $f$ is a retraction, also $g$ can be chosen as retraction.

Figures (6)

  • Figure 1: Three posets $P$ of height two in which the points of $C = \{ a,b,v,w \}$ form a 4-crown in $E(P)$. In the first two posets, $C$ is an improper 4-crown. In the rightmost poset, $C$ is a proper 4-crown, and every edge of every 4-crown in $E(P)$ belongs to an improper 4-crown.
  • Figure 2: The multigraph $\mathfrak{ C }(C)$ with loops omitted in the case of a 4-crown $C$. Explanation in text.
  • Figure 3: Left: A poset $P$ containing two 4-crown bundles $F$ and $G$. Right: The graph $\mathfrak{ C }_{\max}(C)$ for the subposet $C \subset E(P)$ marked by hollow dots in $P$. The lower row of vertices in $\mathfrak{ C }_{\max}(C)$ is from left to right $\vee_C(c)$, $\vee_C(a)$, and $\vee_C(b)$, the upper row is $\wedge_C(v)$, $\wedge_C(w)$, and $\wedge_C(u)$.
  • Figure 4: The poset $N_{a,v}$ contains $m + n - 1$ edges of the up to $m n$ edges of $C$.
  • Figure 5: A poset $P$ containing two 4-crown bundles $F$ and $G$. The subposet $C$ (hollow dots) is a retract and $C_{{\cal F}(P)}$ is a subposet of $N_{b,v}$.
  • ...and 1 more figures

Theorems & Definitions (21)

  • Definition 1
  • Lemma 1
  • proof
  • Proposition 1
  • Lemma 2
  • proof
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • ...and 11 more